$-4-$

$4.(14$ pts$)$ Derive a stiffness$-$to$-$mass performance index for a solid cylindrical shaft that is subjected to a torsional stress. Use the engineering materials listed in the table below and rank these materials based upon their performance for this application. How does this ranking change if including the material cost. $(\backslash (\backslash$rho $\backslash )$ : density, $\backslash ($ E $\backslash )$ : Young's modulus, $\backslash ($ G $\backslash )$ : shear modulus, $\backslash (\backslash$sigma$\_\{\backslash$mathrm$\{$y$\}\}\backslash )$ : yield strength, shear strength $\backslash (=0\mathrm{.}6\backslash$sigma$\_\{$y$\}\backslash )).$

$\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|$c$|$c$|\}$

$\backslash$hline Material & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash (\backslash$rho $\backslash )\backslash \backslash$

$\backslash (\backslash$left$(\backslash$mathrm$\{$Mg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash (\backslash$sigma$\_\{$y$\}\backslash )\backslash \backslash$

$\backslash (($M P a$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash ($ E $\backslash )\backslash \backslash$

$\backslash (($G P a$)\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

$\backslash ($ G $\backslash )\backslash \backslash$

$\backslash ((\backslash$mathrm$\{$GPa$\})\backslash )$

$\backslash$end$\{$tabular$\}$ & $\backslash$begin$\{$tabular$\}\{$c$\}$

Relative $\backslash \backslash$

cost$,\backslash (\backslash$bar$\{$c$\}\backslash )\backslash \backslash$

$\backslash ((\backslash$$ $/\backslash$$$)\backslash )$

$\backslash$end$\{$tabular$\}\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Carbon fiber$-\backslash \backslash$

reinforced composite

$\backslash$end$\{$tabular$\}$ & $1\mathrm{.}5$ & $775$ & $110$ & $9\mathrm{.}2$ & $80\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Stainless steel $304\backslash \backslash$

$($hot finished and $\backslash \backslash$

annealed$)$

$\backslash$end$\{$tabular$\}$ & $8\mathrm{.}0$ & $205$ & $193$ & $74$ & $4\mathrm{.}0\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Aluminum alloy $\backslash \backslash$

$(7075-$T$6)$

$\backslash$end$\{$tabular$\}$ & $2\mathrm{.}8$ & $505$ & $71$ & $26\mathrm{.}7$ & $13\mathrm{.}4\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Magnesium alloy $\backslash \backslash$

$($AZ$31$B extruded$)$

$\backslash$end$\{$tabular$\}$ & $1\mathrm{.}8$ & $220$ & $45$ & $16\mathrm{.}7$ & $12\mathrm{.}6\backslash \backslash$

$\backslash$hline $\backslash$begin$\{$tabular$\}\{$l$\}$

Titanium alloy $($Ti$-\backslash \backslash$

$6$Al$-4$V$)$

$\backslash$end$\{$tabular$\}$ & $4\mathrm{.}4$ & $830$ & $114$ & $42\mathrm{.}5$ & $110\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$